On the Hausdorff dimension of pinned distance sets. (English) Zbl 1414.28011
The paper is a contribution to the conjecture of K. J. Falconer [Mathematika 32, 206–212 (1985; Zbl 0605.28005)] which says that for a set \(A\subseteq \mathbb{R}^d\) with \(d\geq 2\) and Hausdorff dimension \(\dim_H A\geq d/2\) one has \(\dim_H\Delta (A)=1\) where \(\Delta(A):=\{|x-y|:x,y\in A\}\). The author proves that for a set \(A\subseteq\mathbb{R}^2\), for which the Hausdorff and packing dimension coincide and is \(>1\), the set \(\Delta_x(A):=\{|x-y|:y\in A\}\) has full Hausdorff dimension for all \(x\) outside of a set of Hausdorff dimension 1.
Reviewer: Hans Weber (Udine)
Citations:
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